Sound Waves in the Atmosphere. 97 



those results, though of technical importance at the time, 

 are not considered of sufficient general interest for insertion 

 here. 



. V 



§ 1. Summary. 



By an application of a general principle of propagation 

 the equations of propagation of sound waves in a medium in 

 which the velocity and the velocity of sound are given func- 

 tions of positions are obtained in a general form ; they are 

 in two sets, one expressing the convection at each point and 

 the other the refraction. By means of the simplified forms 

 appropriate to a stratified medium such as the atmosphere,, 

 expressions are given for the corrections to the apparent 

 bearing and elevation of a source of sound, and Hill's 

 approximate formulae in terms of the mean wind and 

 temperature lapse are deduced. The range of audibility 

 of an aerial source of sound as limited by total reflexion is 

 then considered ; the conditions for limited range are 

 obtained and curves of extreme audibility are calculated. 

 It is pointed out that if the total reflexion of the boundary 

 ray occurs at an intermediate height and not at the ground, 

 then the boundary ray need not have a zero angle of descent, 

 and the apparent elevation of the source may be non-zero at 

 all ranges. Finally, the conditions for the existence of an 

 envelope to the totally reflected rays are obtained, and the 

 nature of the envelope and of the locus of the vertices of 

 the rays are examined in particular cases. 



§ 2. The Equations of Propagation. 



It seems simplest to deduce the equations of propagation 

 of sound waves in a medium in motion in any manner by 

 assuming the following principle : — 



The motion of a wave-front is the same as if at each 

 moment each point on it were moving with a velocity 

 compounded of (1) the "velocity of sound " at the point 

 considered, taken in the direction of the normal to the 

 wave-front at this point, drawn in the direction in which 

 the wave-front is progressing; (2) the velocity of the 

 medium at the point. 



This principle is sufficient to determine the successive 

 positions of the wave-front, given the motion of the medium 

 and the velocity of sound at each point. A " sound-ray ' is 

 then defined as a curve such that the tangent at each point 

 is in the direction of the resultant velocity mentioned in the 

 Phil. Mag. S. G. Vol. 42. No. 2-17. July 1921. H 



